Which physical system in nonrelativistic quantum mechanics is actually described by a model, where the norm of the "position eigenstate" (i.e. the delta distribution as limit of vectors in the Hilbert state) matters? >What are some actual end-of-calculation quantum mechanical results, which are supposed to mirror some experimental values, where the constant $c$ in the explicit representation of the distribution $c\ \delta(x-y)$, which representas a sharp localization used in calculations, matters? If I write $c$ and I don't set it to 1, how does it enter the result in the end?
If you want to say the norm emerges as a limit of normed states, then justify the norm for the calculation to the point of extracting the specific physical values the theory is supposed to compute.
Also, I guess the "if you would position-localize" statement is a little handwaving, or does this have any practical parallel. (Given that it's not really possible, is it? Especially once relativity comes into play.)
I ask because on the one hand the momentum and position eigenstates/operator are the one you learn first, but in comparison to spins, there bring some mathematical difficulties. And in a quantum mechanics course, most example calculations will be about bounded problems. I guess there might be non-relativistic scattering processes where a delta is actually used to compute some scattering angle. But nonrelativistic scattering is eighter taught in mathematical physic courses, which focus on the structure of the theory (and I don't know practical physical results coming from there), or they are just the results obtained as classical limits of QFT calculations.